A quicker route to the same answer is to reason property-by-property instead of substituting into the gradient formula directly.
The adiabatic gradient \(dT/dr \propto T/C_p\) increases whenever (i) the ambient temperature rises, or (ii) the material's capacity to absorb heat per degree falls. Consider what physically distinguishes the outer core from the lower mantle at the CMB:
Composition change: The lower mantle is a silicate assemblage (dominantly bridgmanite), while the outer core is a liquid iron-nickel alloy. Metals conduct and store thermal energy very differently from silicates -- iron's specific heat capacity is intrinsically lower than that of silicate rock at the same conditions, so crossing into the core lowers \(C_p\).
Thermal state change: Heat flows from the hot core to the cooler mantle, which is only possible if the core is hotter. Geotherm reconstructions consistently show a temperature increase of several hundred kelvin across the CMB thermal boundary layer, confirming \(T_{OC} > T_{LM}\).
Both effects point the same way -- higher \(T\) and lower \(C_p\) in the outer core -- so their combined effect on \(T/C_p\) is amplified, not partially cancelled, which is why the adiabatic gradient steepens sharply across the CMB. Eliminating the other options: (B) and (D) both require \(C_p^{OC} > C_p^{LM}\), which is physically backwards for a metal-versus-silicate contrast; (C) requires the core to be cooler than the mantle, which would reverse the direction of heat flow at the CMB and is inconsistent with a hot, actively convecting core. Only option (A) survives.
\(\boxed{\text{Answer: (A)}}\)